Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom

Transcription

1 Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Ato Thoas S. Kuntzlean Mark Ellison John Tippin Departent of Cheistry Departent of Cheistry Departent of Matheatics Spring Arbor University Ursinus College Spring Arbor University Spring Arbor, MI 4983 P.O. Box Spring Arbor MI 4983 Collegeville, PA 946 Copyright To Kuntzlean, Mark Ellison and John Tippin, 7. All rights reserved. You are welcoe to use this docuent in your own classes but coercial use is not allowed without the perission of the authors. Goal: This docuent illustrates a ethod by which one ay generate and present graphs of the probability distributions of the angular electronic wave function for the hydrogen ato "fro scratch". No attept is ade to derive this wave function. Objectives: After copleting this exercise, students should be able to:. Separately evaluate and graphically display the different angular functions of that are used in constructing the angular electronic wave function for the hydrogen ato, given the quantu nubers l and.. Construct and plot the probability distribution of the angular electronic wave function of the hydrogen ato, given any values of the quantu nubers l and. 3. Describe why the quantu nuber ay only have values that range fro - l to l, according to the associated Legendre polynoial used in constructing the angular electronic wave function of the hydrogen ato. 4. Construct and plot the probability distributions of the angular electronic wave functions of the hydrogen ato that are traditionally displayed for functions having. Introduction: In general, the wavefunction for the angular portion of the electronic wavefunction in the Hydrogen ato is given by: ( Y θ, ) = N P (cosθ) e i

2 Where Y(θ,) is the angular electronic wavefunction of the H-ato, N l is a noralization constant, P l (cosθ) is an associated Legendre polynoial, and e i is an exponential function containing an iaginary coponent. In this docuent we will construct each portion of the angular electronic wavefunction of the H-ato and explore soe of the properties of each portion. Legendre Polynoials: We begin by studying the Legendre polynoials and show how they are obtained and exaine their properties. To find the associated Legendre polynoials, P l (cosθ), we first need to know how to construct the Legendre polynoials, P l (cosθ). The Legendre polynoial is a built-in Mathcad function. For exaple, to call the Legendre polynoial that is nd order in x, you would type Leg(,x). You can observe the characteristics of the first five Legendre polynoials below in Graph. Note carefully the shape of each curve and identify the function type of each line, e.g. linear. Be sure to identify the order of the polynoial with the curve function type. Leg(, x) Leg(, x) Graph :The First 5 Legendre Polynoials Leg(, x) Leg( 3, x) Leg( 4, x) x Alternatively, the l th Laguerre polynoial ay be generated fro the following expression: l l d x Expression l l l! dx

3 The Legendre polynoial which is l th order in x can be generated by substituting the appropriate value for l into the above expression. For exaple, the nd order Legendre polynoial is generated by substituting l = into equation. Notice that the variable l is highlighted in equation (and expression, below) but the nueral is not highlighted. Question : Evaluate expression for l = 5 by replacing each " l" with "5" in expression () above. In the expression for the derivative, replace the l in the lower portion of the derivative expression with a 5 first -- the l in the upper portion will then autoatically be changed as well. Once you have copleted this, highlight the above expression, click on "Sybolics" and then "Siplify". Mathcad should have returned the following polynoial, which is equal to the built-in Mathcad function, Leg(5,x): 63 8 x x x We're going to check to see if we have generated the 5th order Legendre polynoial using expression. Define F(x) and G(x) as the 5th Legendre polynoial as follows: F( x) Leg( 5, x) G( x) 63 8 x x x Here is a plot of both F(x) and G(x): Fx Graph : A coparison of polynoials Gx x

4 The siilarity of these curves strongly suggest F(x) and G(x) are the sae function! Question : a. Use Expression to find the th, st, nd, 3 rd, and 4 th order Legendre polynoial. You should be aware that Mathcad will not evaluate the zeroth derivatives. However, because the zeroth derivative of any function, f(x), is siply f(x), finding the zeroth derivative of a function is trivial. b. What order in x is the st Legendre polynoial? What about the 4 th Legendre polynoial? In general, what order in x is the l th Legendre polynoial? c. Using the Expression and Mathcad's built-in Legendre polynoial function, generate a graph that displays overlay plots of the 3 rd Legendre polynoial generated by these two ethods (See Graph, above). It is actually the associated Legendre polynoials that are used in the constructing the P l (cosθ) portion of the angular electronic orbitals of the H-ato. The th associated Legendre polynoial of order l is generated using Expression : ( ) l l! ( x ) l+ d l+ dx ( x ) l Expression You ay have quessed that the values of l and which are substituted into Expression are the values for l and of the atoic orbital you wish to contruct (where l is the orbital angular oentu quantu nuber and is the agnetic quantu nuber). Recall that for any value of l, can take on values fro -l to l. Let's use Expression to find the associated Legendre polynoial that is used to construct part of a p + orbital. To do this, we substitute l = and = into expression above. You ay want to copy and paste expression into the space below. Notice again, that l and are higlighted in the above expression, but nuerals are not. Then highlight this expression you have generated, choose "Sybolics", "Siplify" fro the toolbar. If you get -(-x ) / as your answer, you have evaluated the the expression correctly. Question 3: a. Find the nd associated Legendre polynoial of order, and the nd associated Legendre polynoial of order 3. b. Use expression to find the second Legendre polynoial, P (x), by setting =. You should note that when =, the associated Legendre polynoial reduces to the Legendre polynoial

5 Legendre polynoial. c. Carefully exaine expression. If =, what order polynoial will be generated for a given value of l? c. Again, carefully exaine expression. What order polynoial will be generated if l =? What order polynoial will be generated if > l? For a given value of l, what is the axiu value of that will give a non-zero associated Legendre polynoial? d. According to your answer in part c, why ust is be that l? Use this fact to explain why the values of the quantu nuber are allowed to vary fro -l to l. Finally, we notice that the associated Legendre polynoial is defined in ters of θ, which is the angle of displaceent fro the vertical (z) axis. Specifically, the associated Legendre polynoial is defined as P l (cosθ). To construct the associated Legendre polynoial in this for, we siply substitute cosθ = x into the appropriate associated Legendre polynoial. Because x = cosθ and because θ varies fro to π, we note that cos() = and cos(π) = -. Thus, the for of an associated Legendre polynoial, P l (cosθ), fro to π should be siilar in for to the associated Legendre polynoial, P l (x), fro to -. Let's explore this siilarity in a bit ore detail. First, define l = and = (to what type of orbital does this correspond?): l Now find the appropriate associated Legendre polynoial, by substituting the appropriate values for l and into expression : and sybolically evaluating: ( )! ( x ) + d dx + ( x ) Did you find this polynoial to be equal to 3 x x? If not, try again until you do. Define this as P(x) and PC(θ), aking the substitution x = cosθ for the latter. Px 3 x x 3 ( cos( θ) ) PC θ cos( θ) We now graph P(x) vs. x (Graph 3a) and PC(θ) vs. cos(θ) (Graph 3b).

6 Graph 3a Graph 3b ( Px ) PC( θ) x cos( θ) Question 4: a. To what type of orbital does l = and = correspond? b. Generate graphs siilar to those of Graph 3a and 3b to copare P(x) to PC( θ) for any s orbital. Repeat for any p orbital. The -dependent Function of the Angular Electronic Wave Function: We now turn to the -dependent portion of the angular electronic wavefunction of the H-ato, which is quite easy to construct in Mathcad. The for of this function is given below: Φ = e π i Recall that possible values for =,,... for the solutions to the Φ portion of the angular wavefunction. We'll define = below: Now we define Φ(): Φ π e i Although Φ is not real when, we can get soe idea of the behavior of this function by graphing its real and iaginary coponents separately:

7 .5 Graph 4a.5 Graph 4b Re( Φ ) I( Φ ) Question 5: a. Why is Φ() real when =? b. Φ() is an exponential function of, but the real and iaginary portions appear to vary cosinusoidally and sinusoidally, respectively. Why? HINT: Recall Euler's theore. c. What changes do you observe in the real and iaginary portions of Φ() when you change the value of? d. What would you expect the probability distribution of Φ() to look like? Why? The Noralization Constant: Finally, we turn to the noralization constant, N l,. N l, depends upon l and, we need to define these nubers (Keep in ind you need to define these appropriately so that the associated Legendre polynoial does not vanish (see questions 3c and 3d, above): The noralization constant is: l N ( l + ) ( l )! ( l + )! N =.5 Let's check to see of PC(θ) is noralized: 3 ( cos( θ) ) PC θ cos( θ)

8 π PC θ PC ( θ ) sin( θ) dθ =.4 The function is not noralized. We need to ake the wavefunction noralized. (Why is it iportant to have the wavefunction noralized?) If we include the noralization constant, is this function noralized? π NPCθ NPC θ sin( θ) dθ = 3.6 Question 6: Check that PC(θ) is noralized for l=, =. Construction of the Entire Wave Function and Probability Distribution: We can now construct the full angular electronic wavefunction for the H-ato. Let's do this for a p z orbital, for which l = and =. We first define l and below: l The noralization constant is then: N ( l + ) ( l )! ( l + )! N =.5 The appropriate associated Legendre polynoial is defined as P(, l,x), using expression : ( ) P(, l, x) l l! ( x ) l+ d l+ dx ( x ) l

9 : Now we can put it all together in the angular wavefunction, which we define as Ψ θ,, l, l + Ψ θ,, l, ( l )! P (, l, cos( θ) ) ( exp( i) ) Equation 3 4π ( l + )! where l and are the failiar quantu nubers, θ is the angle displaced fro the z-axis, and is an angle that is swept out within the xy plane. The first factor of Equation 3, (l+)(l- )! / 4π(l+ )! is the noralization constant for these orbitals. The second factor, P(,l,cos(θ)), is the associated Legendre polynoial in θ. The third factor, exp(i), represents the portion of the wavefunction dependent upon. Notice that this portion of the wavefunction will yield iaginary results for any value of >. However, because we are only interested in the probability distribution of this wavefunction, we will square Ψ(θ,,l,) (ultiply by its coplex conjugate) or square linear cobinations of wavefunctions (ultiply the by their coplex conjugates) with the sae eignvalues as Ψ(θ,,l,). Let's construct and display the probability distribution of the angular portion of the p orbital. To do so, we have to construct each of the three parts of Equation 3: The noralization constant, the associated Legendre polynoial transfored into a function of θ, and the function of. We define l and for a p orbital: l NOTE: Define HERE the values for l and that you want to use for the probability plots (see below).. First, we'll deterine the noralization constant: N ( l + ) ( l )! 4π ( l + )! Next, we need to deterine the associated Legendre polynoial that is part of the angular part of the wavefunction. This is Expression fro above, which allows us to deterine the associated Legendre polynoial for specific values of the quantu nubers l and. Evaluate Expression sybolically for the proper values of the l and quantu nubers for a p orbital.

10 ( ) l l! ( x ) l l+ d x Expression l+ dx Did you find the value to be x? Good! Now be sure to cut and paste your answer into the space at the right hand side of the definition below to define P(x): P( x) x NOTE: Define HERE the associated Legendre polynoial that you want to use for the probability plots (see below). siply cut and paste fro your siplified expression (above). Later on, we'll ake the substitution, x = cosθ to transfor P(x) to P(cos(θ)). This is siply done by writing P(cos(θ)), which will allow Mathcad to do the substitution for us. This will coe in handy with soe of the ore coplicated Legendre polynoials. In the case for l = and =, however, you should note that P(cos(θ)) = cos(θ). Now we define the function for Φ(): Φ π e i Now that we have each portion angular wavefunction defined, we can define Ψ(θ,) as a product of N, P(cos(θ)), and Φ(): NP( cos( θ) ) Ψ θ, Φ Multiplying each wavefunction by its coplex conjugate and setting that equal to r allows us to visualize the angular part of the wavefunction in three diensions. Here, r is not the distance fro the nucleus. Rather, it is related to the probability of finding the electron at a location specified by the angles (θ,) in space. To display the probability distribution of the wavefunction, we will ultiply Ψ(θ,) by its coplex conjugate. In addition, the substitutions x = r sinθ cos, y = r sinθ sin and z = r cosθ are also ade, where r = Ψ Ψ. The transforation of variables is done so we can display the wavefunction in Cartesian coordinates. Denoting the coplex conjugate of a function is done in Mathcad by placing a bar over the function. This is done by underlining the function you wish to take the coplex conjugate of and typing a quote ("), or (Shift ' ).

11 Ψ( θ, ) Ψ ( θ, ) X θ, Y θ, ( ) sin ( θ ) ( ) sin ( θ ) Ψ( θ, ) Ψ ( θ, ) Ψ( θ, ) Ψ ( θ, ) Z θ, cos θ cos sin Now display the angular wavefunction. First, click on Insert, Graph, then Surface Plot (Alternatively, hit Ctrl ). There should be a black rectangle at the botto left-hand corner of the graph. Type (X, Y, Z) in this rectangle (include the parentheses!). Graph 5: Display of the Angular Portion of the Po orbital ( X, Y, Z)

12 Question 7: a. Display the probability distribution for a s orbital above. Go back to expression and evaluate the appropriate associated Legendre polynoial for a s orbital ( l =, = ). (Be careful! Mathcad won't calculate a zeroth derivative! However, the "zeroth derivative" of any function is siply the function itself!) Define this polynoial as P(x) in yellow above. Do you get the failar result? b. Display the probability distribution for the 3d z orbital above. Go back to expression and evaluate the appropriate associated Legendre polynoial for a 3d z orbital (l =, = ). Define this polynoial as P(x) in yellow above. Do you get the failar result? c. Display the probability distribution for the p + and p - orbitals above. Go back to expression and evaluate the appropriate associated Legendre polynoial for a p + and p - orbitals (l =, = or -). Define this polynoial as P(x) in yellow above. Do you get the failar result? You probably noted in part c of question 4 that the probability distributions of the p + and p - are not those norally displayed in text books. Custoarily, linear cobinations of eigenfunctions of the Φ() portion of the angular electronic wavefunction of the H-ato are used in the construction of the probability distributions when. This is done for two reasons. First, when, the angular electronic wavefunction not only depends upon both θ and, but also contains an iaginary coponent in the dependence. Second, the probability distributions for the = and = - (or any other cobination of = plus or inus x) wavefunctions are identical. But cheists are very interested in the directional character of each orbital, as this offers clues into how atos interact in bonding. So how are these "textbook" probability distributions graphed? Linear cobinations of eigenfunctions of the Φ() portions are constructed. The -dependent wavefunctions, Φ() px and Φ() py, are eigenfunctions of the Schrodinger equation for the -dependent portion of the H-ato electron. Because quantu operators are linear operators, linear cobinations of these eigenfunctions ust also be eigenfunctions. By taking linear cobinations of the Φ() p+ andφ() p- wavefunctions and ultipling these linear cobinations (different ones for each case) with N l, and P l (cosθ), we can construct the "texbook" p + and p - wave functions and probability distributions. To display the Φ portion of the p (l =, = -,, ) angular orbital wavefunctions, we define: Φinus e i ( ) for = - Φzero e i ( ) for = Φplus e i for = +

13 Clearly, Φ(), Φp() and higher are evaluated by siply substituting in the appropriate value for : Φinus e i ( ) for = - Φplus e i ( ) for = + At any rate, real representations of the -dependent portion of p +, p - and p orbitals are then constructed by the following linear cobinations: Φx Φy Φplus + Φinus i( Φplus Φinus ) e i Φz ( ) We view these linear cobinations to verify that they are indeed real: Graph 6a Graph 6b Φx Φy

14 . Graph 6c Φz Viewing the real representations of the p + and p - orbitals is now easy! For p +, l = and =. We therefore define l, and deterine the noralization constant: l N ( l + ) ( l )! 4π ( l + )! Now we use Expression to find and define the appropriate Legendre polynoial: ( )! ( x ) + d dx + ( x ) P( x) x ( x ) We redefine Ψ(Θ,) to include the real representation of the -dependent portion (Φx()) for the p x orbital: NP( cos( θ) ) Ψ θ, Φx

15 Since we want the probability distribution, we ultiply each function above by its coplex conjugate, take each product and transfor into Cartesian coordinates: Ψ( θ, ) Ψ ( θ, ) X θ, Y θ, ( ) sin ( θ ) ( ) sin ( θ ) Ψ( θ, ) Ψ ( θ, ) Ψ( θ, ) Ψ ( θ, ) Z θ, cos θ cos sin Finally, we display the orbital: Graph 7: Display of atoic orbitals ( X, Y, Z) Question 8: a. Does the orbital in graph 7 represent the "textbook" version of a p x orbital? b. How does the orientation of the orbital in graph 5 copare the the orientation of the orbital in graph 7, with respect to the x, y and z-axes?

16 Question 9: a. Display the probability distribution of the "textbook version" the p y (l =, = -) orbital in Cartesian coordinates. b. The "textbook" representations of the 3d orbitals are constructed by using the fro the following linear cobinations in the -dependent portions of the angular electronic wavefunction of the H-ato: d z = Φ() (l =, = ) d xz = [Φplus() + Φinus()] / (l =, = ) d yz = -i[φplus() - Φinus()] / (l =, = -) d x -y = [Φplus() + Φinus()] / (l =, = ) d xy = -i[φplus() + Φinus()] / (l =, = -) Define equations for, and display in spherical coordinates the: a. d z orbital b. two other 3d orbitals Mastery Exercise: Display the "textbook" wavefunctions and probability distributions for a few of the 7 f orbitals. You ay need to do a bit of literature or internet searching to find the appropriate linear cobinations of eigenfunctions to use in the construction of these probability distribution plots.